I wonder if with current day technology we could make a "floating platform" which hangs from a satellite by a space elevator. This could allow a "launch process" involving floating a balloon to the upper atmosphere, then transferring the payload to the elevator to complete the travel to space.
If the cable of the "elevator" is not connected to a point on earth, then the satellite must be in a geostationary orbit (or it will float away); this implies that if you now attach something to the platform (increasing the pull on the cable) you will pull the satellite down to earth.
And as @lionelbrits pointed out, the pulling part of a space elevator would have to be beyond geostationary orbit in order to provide a net pull - and this means the cable has to be just about as long as if you tethered it to the ground (but the tethered one would work...).
Another way to think about this - there has to be a net tension in the cable at the point where you load it, otherwise the elevator won't work.
I've read a little bit on the science of a space elevator and it's a surprisingly difficult problem. To have a working space elevator, it would need to be at least to the Geosynchronous orbit, 22,000 miles up, probobly a bit beyond that for buoyancy. The highest balloon is some 25 miles - so that's less than 1/10th of 1% of the distance.
The strongest tether that can currently be manufactured that length is Kevlar and the weight of a 22,000 mile length of Kevlar is too much for it's strength to bear. A rough calculation (based on Kevlar rope sold on Amazon) 100 feet of Kevlar rope (1/8th inch thickness, 900 lbs tensil strength), weighs about 4 OZ, so 1 mile, weight is about 200 OZ or 12 lbs, 22,000 miles - 260,000 lbs, now, the earth's gravity decreases with distance, so, the actual mass might be closer to 100,000 lbs, but that's for a thin rope that has a listed strength of 900 lbs. 100 times to weak. I'm sure there's better products out there, but the best Kevlar or steel wire rope isn't nearly strong enough for what a space elevator rope or tether would need to do. My calculation is rough, granted, but it's generally true. Almost nothing is strong enough to be made into a space tether, used as an elevator. (footnote, I've read that the best Kevlar tether is about 1/10th as strong as it needs to be, but that's still a long way from good enough).
Carbon nano-tube fiber is the only substance that should be strong enough but it's difficult to build a strand that long. There's other problems, like a space tether would be pass through satellite orbit, so it would need to avoid being crashed into. It would need to withstand radiation from space and weather on earth and perhaps the very occasional meteor strike, so there's stuff to consider like what would happen if it breaks.
To have the proper tension, you'd need an anchor on both sides, 1 orbiting in space, presumably a bit beyond the geosynchronous orbit and on on earth, presumably floating in the ocean which would enable more mobility. There's no real benefit to building just a space elevator without a space platform in that orbit attached to it, so it's a pretty enormous project. It wouldn't be possible, for example, to run a space elevator to the current space station because that orbits the earth every 1 1/2 hours. it's moving way too fast. You'd need a much higher orbit to correspond with earth's rotation.
Curious fact regarding the moon, but a Kevlar tether would be strong enough for a space elevator on the moon.